\(\int \frac {\coth ^6(x)}{(i+\sinh (x))^2} \, dx\) [227]

Optimal result

Integrand size = 13, antiderivative size = 48 \[ \int \frac {\coth ^6(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{4} i \text {arctanh}(\cosh (x))-\frac {2 \coth ^3(x)}{3}+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth (x) \text {csch}(x)+\frac {1}{2} i \coth (x) \text {csch}^3(x) \]

[Out]

-1/4*I*arctanh(cosh(x))-2/3*coth(x)^3+1/5*coth(x)^5+1/4*I*coth(x)*csch(x)+1/2*I*coth(x)*csch(x)^3

Rubi [A] (verified)

Time = 0.06 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {2788, 3852, 8, 3853, 3855} \[ \int \frac {\coth ^6(x)}{(i+\sinh (x))^2} \, dx=-\frac {1}{4} i \text {arctanh}(\cosh (x))+\frac {\coth ^5(x)}{5}-\frac {2 \coth ^3(x)}{3}+\frac {1}{2} i \coth (x) \text {csch}^3(x)+\frac {1}{4} i \coth (x) \text {csch}(x) \]

[In]

Int[Coth[x]^6/(I + Sinh[x])^2,x]

[Out]

(-1/4*I)*ArcTanh[Cosh[x]] - (2*Coth[x]^3)/3 + Coth[x]^5/5 + (I/4)*Coth[x]*Csch[x] + (I/2)*Coth[x]*Csch[x]^3

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 2788

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*tan[(e_.) + (f_.)*(x_)]^(p_), x_Symbol] :> Dist[a^p, Int[Expan
dIntegrand[Sin[e + f*x]^p*((a + b*Sin[e + f*x])^(m - p/2)/(a - b*Sin[e + f*x])^(p/2)), x], x], x] /; FreeQ[{a,
b, e, f}, x] && EqQ[a^2 - b^2, 0] && IntegersQ[m, p/2] && (LtQ[p, 0] || GtQ[m - p/2, 0])

Rule 3852

Int[csc[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Dist[-d^(-1), Subst[Int[ExpandIntegrand[(1 + x^2)^(n/2 - 1), x]
, x], x, Cot[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[n/2, 0]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \int \left (\text {csch}^2(x)-2 i \text {csch}^3(x)-2 i \text {csch}^5(x)-\text {csch}^6(x)\right ) \, dx \\ & = -\left (2 i \int \text {csch}^3(x) \, dx\right )-2 i \int \text {csch}^5(x) \, dx+\int \text {csch}^2(x) \, dx-\int \text {csch}^6(x) \, dx \\ & = i \coth (x) \text {csch}(x)+\frac {1}{2} i \coth (x) \text {csch}^3(x)+i \int \text {csch}(x) \, dx-i \text {Subst}(\int 1 \, dx,x,-i \coth (x))+i \text {Subst}\left (\int \left (1+2 x^2+x^4\right ) \, dx,x,-i \coth (x)\right )+\frac {3}{2} i \int \text {csch}^3(x) \, dx \\ & = -i \text {arctanh}(\cosh (x))-\frac {2 \coth ^3(x)}{3}+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth (x) \text {csch}(x)+\frac {1}{2} i \coth (x) \text {csch}^3(x)-\frac {3}{4} i \int \text {csch}(x) \, dx \\ & = -\frac {1}{4} i \text {arctanh}(\cosh (x))-\frac {2 \coth ^3(x)}{3}+\frac {\coth ^5(x)}{5}+\frac {1}{4} i \coth (x) \text {csch}(x)+\frac {1}{2} i \coth (x) \text {csch}^3(x) \\ \end{align*}

Mathematica [B] (verified)

Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(175\) vs. \(2(48)=96\).

Time = 0.11 (sec) , antiderivative size = 175, normalized size of antiderivative = 3.65 \[ \int \frac {\coth ^6(x)}{(i+\sinh (x))^2} \, dx=-\frac {7}{30} \coth \left (\frac {x}{2}\right )+\frac {1}{16} i \text {csch}^2\left (\frac {x}{2}\right )-\frac {19}{480} \coth \left (\frac {x}{2}\right ) \text {csch}^2\left (\frac {x}{2}\right )+\frac {1}{32} i \text {csch}^4\left (\frac {x}{2}\right )+\frac {1}{160} \coth \left (\frac {x}{2}\right ) \text {csch}^4\left (\frac {x}{2}\right )-\frac {1}{4} i \log \left (\cosh \left (\frac {x}{2}\right )\right )+\frac {1}{4} i \log \left (\sinh \left (\frac {x}{2}\right )\right )+\frac {1}{16} i \text {sech}^2\left (\frac {x}{2}\right )-\frac {1}{32} i \text {sech}^4\left (\frac {x}{2}\right )-\frac {7}{30} \tanh \left (\frac {x}{2}\right )+\frac {19}{480} \text {sech}^2\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right )+\frac {1}{160} \text {sech}^4\left (\frac {x}{2}\right ) \tanh \left (\frac {x}{2}\right ) \]

[In]

Integrate[Coth[x]^6/(I + Sinh[x])^2,x]

[Out]

(-7*Coth[x/2])/30 + (I/16)*Csch[x/2]^2 - (19*Coth[x/2]*Csch[x/2]^2)/480 + (I/32)*Csch[x/2]^4 + (Coth[x/2]*Csch
[x/2]^4)/160 - (I/4)*Log[Cosh[x/2]] + (I/4)*Log[Sinh[x/2]] + (I/16)*Sech[x/2]^2 - (I/32)*Sech[x/2]^4 - (7*Tanh
[x/2])/30 + (19*Sech[x/2]^2*Tanh[x/2])/480 + (Sech[x/2]^4*Tanh[x/2])/160

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 73 vs. \(2 (35 ) = 70\).

Time = 61.78 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.54

method	result	size
default	\(-\frac {3 \tanh \left (\frac {x}{2}\right )}{16}+\frac {\tanh \left (\frac {x}{2}\right )^{5}}{160}-\frac {i \tanh \left (\frac {x}{2}\right )^{4}}{32}-\frac {5 \tanh \left (\frac {x}{2}\right )^{3}}{96}+\frac {i}{32 \tanh \left (\frac {x}{2}\right )^{4}}+\frac {1}{160 \tanh \left (\frac {x}{2}\right )^{5}}-\frac {3}{16 \tanh \left (\frac {x}{2}\right )}-\frac {5}{96 \tanh \left (\frac {x}{2}\right )^{3}}+\frac {i \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{4}\)	\(74\)
risch	\(\frac {i \left (60 i {\mathrm e}^{8 x}+15 \,{\mathrm e}^{9 x}-240 i {\mathrm e}^{6 x}+90 \,{\mathrm e}^{7 x}+40 i {\mathrm e}^{4 x}-80 i {\mathrm e}^{2 x}-90 \,{\mathrm e}^{3 x}+28 i-15 \,{\mathrm e}^{x}\right )}{30 \left ({\mathrm e}^{2 x}-1\right )^{5}}-\frac {i \ln \left ({\mathrm e}^{x}+1\right )}{4}+\frac {i \ln \left ({\mathrm e}^{x}-1\right )}{4}\)	\(82\)

[In]

int(coth(x)^6/(I+sinh(x))^2,x,method=_RETURNVERBOSE)

[Out]

-3/16*tanh(1/2*x)+1/160*tanh(1/2*x)^5-1/32*I*tanh(1/2*x)^4-5/96*tanh(1/2*x)^3+1/32*I/tanh(1/2*x)^4+1/160/tanh(
1/2*x)^5-3/16/tanh(1/2*x)-5/96/tanh(1/2*x)^3+1/4*I*ln(tanh(1/2*x))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 160 vs. \(2 (32) = 64\).

Time = 0.28 (sec) , antiderivative size = 160, normalized size of antiderivative = 3.33 \[ \int \frac {\coth ^6(x)}{(i+\sinh (x))^2} \, dx=-\frac {15 \, {\left (i \, e^{\left (10 \, x\right )} - 5 i \, e^{\left (8 \, x\right )} + 10 i \, e^{\left (6 \, x\right )} - 10 i \, e^{\left (4 \, x\right )} + 5 i \, e^{\left (2 \, x\right )} - i\right )} \log \left (e^{x} + 1\right ) + 15 \, {\left (-i \, e^{\left (10 \, x\right )} + 5 i \, e^{\left (8 \, x\right )} - 10 i \, e^{\left (6 \, x\right )} + 10 i \, e^{\left (4 \, x\right )} - 5 i \, e^{\left (2 \, x\right )} + i\right )} \log \left (e^{x} - 1\right ) - 30 i \, e^{\left (9 \, x\right )} + 120 \, e^{\left (8 \, x\right )} - 180 i \, e^{\left (7 \, x\right )} - 480 \, e^{\left (6 \, x\right )} + 80 \, e^{\left (4 \, x\right )} + 180 i \, e^{\left (3 \, x\right )} - 160 \, e^{\left (2 \, x\right )} + 30 i \, e^{x} + 56}{60 \, {\left (e^{\left (10 \, x\right )} - 5 \, e^{\left (8 \, x\right )} + 10 \, e^{\left (6 \, x\right )} - 10 \, e^{\left (4 \, x\right )} + 5 \, e^{\left (2 \, x\right )} - 1\right )}} \]

[In]

integrate(coth(x)^6/(I+sinh(x))^2,x, algorithm="fricas")

[Out]

-1/60*(15*(I*e^(10*x) - 5*I*e^(8*x) + 10*I*e^(6*x) - 10*I*e^(4*x) + 5*I*e^(2*x) - I)*log(e^x + 1) + 15*(-I*e^(
10*x) + 5*I*e^(8*x) - 10*I*e^(6*x) + 10*I*e^(4*x) - 5*I*e^(2*x) + I)*log(e^x - 1) - 30*I*e^(9*x) + 120*e^(8*x)
- 180*I*e^(7*x) - 480*e^(6*x) + 80*e^(4*x) + 180*I*e^(3*x) - 160*e^(2*x) + 30*I*e^x + 56)/(e^(10*x) - 5*e^(8*
x) + 10*e^(6*x) - 10*e^(4*x) + 5*e^(2*x) - 1)

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 114 vs. \(2 (44) = 88\).

Time = 0.16 (sec) , antiderivative size = 114, normalized size of antiderivative = 2.38 \[ \int \frac {\coth ^6(x)}{(i+\sinh (x))^2} \, dx=\operatorname {RootSum} {\left (16 z^{2} + 1, \left ( i \mapsto i \log {\left (4 i i + e^{x} \right )} \right )\right )} + \frac {15 i e^{9 x} - 60 e^{8 x} + 90 i e^{7 x} + 240 e^{6 x} - 40 e^{4 x} - 90 i e^{3 x} + 80 e^{2 x} - 15 i e^{x} - 28}{30 e^{10 x} - 150 e^{8 x} + 300 e^{6 x} - 300 e^{4 x} + 150 e^{2 x} - 30} \]

[In]

integrate(coth(x)**6/(I+sinh(x))**2,x)

[Out]

RootSum(16*_z**2 + 1, Lambda(_i, _i*log(4*_i*I + exp(x)))) + (15*I*exp(9*x) - 60*exp(8*x) + 90*I*exp(7*x) + 24
0*exp(6*x) - 40*exp(4*x) - 90*I*exp(3*x) + 80*exp(2*x) - 15*I*exp(x) - 28)/(30*exp(10*x) - 150*exp(8*x) + 300*
exp(6*x) - 300*exp(4*x) + 150*exp(2*x) - 30)

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 103 vs. \(2 (32) = 64\).

Time = 0.22 (sec) , antiderivative size = 103, normalized size of antiderivative = 2.15 \[ \int \frac {\coth ^6(x)}{(i+\sinh (x))^2} \, dx=\frac {-15 i \, e^{\left (-x\right )} - 80 \, e^{\left (-2 \, x\right )} - 90 i \, e^{\left (-3 \, x\right )} + 40 \, e^{\left (-4 \, x\right )} - 240 \, e^{\left (-6 \, x\right )} + 90 i \, e^{\left (-7 \, x\right )} + 60 \, e^{\left (-8 \, x\right )} + 15 i \, e^{\left (-9 \, x\right )} + 28}{30 \, {\left (5 \, e^{\left (-2 \, x\right )} - 10 \, e^{\left (-4 \, x\right )} + 10 \, e^{\left (-6 \, x\right )} - 5 \, e^{\left (-8 \, x\right )} + e^{\left (-10 \, x\right )} - 1\right )}} - \frac {1}{4} i \, \log \left (e^{\left (-x\right )} + 1\right ) + \frac {1}{4} i \, \log \left (e^{\left (-x\right )} - 1\right ) \]

[In]

integrate(coth(x)^6/(I+sinh(x))^2,x, algorithm="maxima")

[Out]

1/30*(-15*I*e^(-x) - 80*e^(-2*x) - 90*I*e^(-3*x) + 40*e^(-4*x) - 240*e^(-6*x) + 90*I*e^(-7*x) + 60*e^(-8*x) +
15*I*e^(-9*x) + 28)/(5*e^(-2*x) - 10*e^(-4*x) + 10*e^(-6*x) - 5*e^(-8*x) + e^(-10*x) - 1) - 1/4*I*log(e^(-x) +
1) + 1/4*I*log(e^(-x) - 1)

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 74 vs. \(2 (32) = 64\).

Time = 0.27 (sec) , antiderivative size = 74, normalized size of antiderivative = 1.54 \[ \int \frac {\coth ^6(x)}{(i+\sinh (x))^2} \, dx=-\frac {-15 i \, e^{\left (9 \, x\right )} + 60 \, e^{\left (8 \, x\right )} - 90 i \, e^{\left (7 \, x\right )} - 240 \, e^{\left (6 \, x\right )} + 40 \, e^{\left (4 \, x\right )} + 90 i \, e^{\left (3 \, x\right )} - 80 \, e^{\left (2 \, x\right )} + 15 i \, e^{x} + 28}{30 \, {\left (e^{\left (2 \, x\right )} - 1\right )}^{5}} - \frac {1}{4} i \, \log \left (e^{x} + 1\right ) + \frac {1}{4} i \, \log \left ({\left | e^{x} - 1 \right |}\right ) \]

[In]

integrate(coth(x)^6/(I+sinh(x))^2,x, algorithm="giac")

[Out]

-1/30*(-15*I*e^(9*x) + 60*e^(8*x) - 90*I*e^(7*x) - 240*e^(6*x) + 40*e^(4*x) + 90*I*e^(3*x) - 80*e^(2*x) + 15*I
*e^x + 28)/(e^(2*x) - 1)^5 - 1/4*I*log(e^x + 1) + 1/4*I*log(abs(e^x - 1))

Mupad [B] (verification not implemented)

Time = 1.51 (sec) , antiderivative size = 246, normalized size of antiderivative = 5.12 \[ \int \frac {\coth ^6(x)}{(i+\sinh (x))^2} \, dx=-\frac {80\,{\mathrm {e}}^{4\,x}-160\,{\mathrm {e}}^{2\,x}-480\,{\mathrm {e}}^{6\,x}+120\,{\mathrm {e}}^{8\,x}+56-\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,15{}\mathrm {i}+\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,15{}\mathrm {i}+{\mathrm {e}}^{3\,x}\,180{}\mathrm {i}-{\mathrm {e}}^{7\,x}\,180{}\mathrm {i}-{\mathrm {e}}^{9\,x}\,30{}\mathrm {i}+{\mathrm {e}}^x\,30{}\mathrm {i}+\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{2\,x}\,75{}\mathrm {i}-\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{2\,x}\,75{}\mathrm {i}-\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{4\,x}\,150{}\mathrm {i}+\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{4\,x}\,150{}\mathrm {i}+\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{6\,x}\,150{}\mathrm {i}-\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{6\,x}\,150{}\mathrm {i}-\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{8\,x}\,75{}\mathrm {i}+\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{8\,x}\,75{}\mathrm {i}+\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}-\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{10\,x}\,15{}\mathrm {i}-\ln \left (-\frac {{\mathrm {e}}^x\,1{}\mathrm {i}}{2}+\frac {1}{2}{}\mathrm {i}\right )\,{\mathrm {e}}^{10\,x}\,15{}\mathrm {i}}{60\,{\left ({\mathrm {e}}^{2\,x}-1\right )}^5} \]

[In]

int(coth(x)^6/(sinh(x) + 1i)^2,x)

[Out]

-(log(1i/2 - (exp(x)*1i)/2)*15i - log(- (exp(x)*1i)/2 - 1i/2)*15i - 160*exp(2*x) + exp(3*x)*180i + 80*exp(4*x)
- 480*exp(6*x) - exp(7*x)*180i + 120*exp(8*x) - exp(9*x)*30i + exp(x)*30i + log(- (exp(x)*1i)/2 - 1i/2)*exp(2
*x)*75i - log(1i/2 - (exp(x)*1i)/2)*exp(2*x)*75i - log(- (exp(x)*1i)/2 - 1i/2)*exp(4*x)*150i + log(1i/2 - (exp
(x)*1i)/2)*exp(4*x)*150i + log(- (exp(x)*1i)/2 - 1i/2)*exp(6*x)*150i - log(1i/2 - (exp(x)*1i)/2)*exp(6*x)*150i
- log(- (exp(x)*1i)/2 - 1i/2)*exp(8*x)*75i + log(1i/2 - (exp(x)*1i)/2)*exp(8*x)*75i + log(- (exp(x)*1i)/2 - 1
i/2)*exp(10*x)*15i - log(1i/2 - (exp(x)*1i)/2)*exp(10*x)*15i + 56)/(60*(exp(2*x) - 1)^5)